Spectrum of one dimensional p-Laplacian Operator with indefinite weight

A. Anane^{1}, O. Chakrone^{1} and M. Moussa^{2}

1Département de mathématiques, Faculté des Sciences, Université Mohamed I^{er}, Oujda. Maroc. e-mail: anane@sciences.univ-oujda.ac.ma chakrone@sciences.univ-oujda.ac.ma 2Départment de mathématiques, Faculté des Sciences, Université Ibn Tofail, Kénitra. Maroc. e-mail: moussa@univ-ibntofail.ac.ma

Abstract

This paper is concerned with the nonlinear boundary eigenvalue problem

−(|u′|^{p−2}u′)′ = λm|u|^{p−2}u u ∈ I =]a, b[, u(a) = u(b) = 0,

where p > 1, λ is a real parameter, m is an indefinite weight, and a, b are real numbers. We prove there exists a unique sequence of eigenvalues for this problem. Each eigenvalue is simple and verifies the strict monotonicity property with respect to the weight m and the domain I, the k-th eigenfunction, corresponding to the k-th eigenvalue, has exactly k − 1 zeros in (a, b). At the end, we give a simple variational formulation of eigenvalues. MSC 2000 35P30. Key words and phrases: p-Laplacian spectrum, simplicity, isolation, strict monotonicity property, zeros of eigenfunctions, variational formulation.

1 Introduction

The spectrum of the p-Laplacian operator with indefinite weight is defined as the set σ_{p}(−Δ_{p}, m) of λ := λ(m, I) for which there exists a nontrivial (weak) solution u ∈ W_{0}^{1,p}(Ω) of problem

(V.P_{(m,Ω)}) \{ \begin{matrix}−Δ_{p}u & = & λm|u|^{p−2}u & in & Ω, \\ u & = & 0 & on & ∂Ω, \end{matrix}

where p > 1, Δ_{p} : is the p-Laplacian operator, defined by Δ_{p}u := div(|∇u|^{p−2}∇u), in a bounded domain Ω ⊂ IR^{N}, and m ∈ M(Ω) is an indefinite weight, with

M(Ω) := {m ∈ L^{∞}(Ω)/meas{x ∈ Ω, m(x) > 0} ≠ 0}.

The values λ(m, Ω) for which there exists a nontrivial solution of (V.P_{(m,Ω)}) are called eigenvalues and the corresponding solutions are the eigenfunctions. We will denote σ_{p}^{+}(−Δ_{p}, m) the set of all positive eigenvalues, and by σ_{p}^{−}(−Δ_{p}, m) the set of negative eigenvalues.

For p = 2 (Δ_{p} = Δ Laplacian operator) it is well known (cf [4, 7, 8]) that,

• σ_{2}^{+}(−Δ, m) = {μ_{k}(m, Ω), k = 1, 2, ⋯}, with 0 < μ_{1}(m, Ω) < μ_{2}(m, Ω) ≤ μ_{3}(m, Ω) ≤ ⋯ → +∞, μ_{k}(m, Ω) repeated according to its multiplicity.

• The k-th eigenfunction corresponding to μ_{k}(m, Ω), has at most k nodal domains.

• The eigenvalues μ_{k}(m, Ω), k ≥ 1, verify the strict monotonicity property (SMP in brief), i.e if m, m′ ∈ M(Ω), m(x) ≤ m′(x) a.e in Ω and m(x) < m′(x) in some subset of nonzero measure, then μ_{k}(m, Ω) > μ_{k}(m′, Ω).

• Equivalence between the SMP and the unique continuation one.

For p ≠ 2 (nonlinear problem), it is well known that the critical point theory of Ljusternik-Schnirelmann (cf [15]), provides a sequence of eigenvalues for those problems. Whether or not this sequence, denoted λ_{k}(m, Ω), constitutes the set of all eigenvalues is an open question when N ≥ 1, m ≢ 1, and p ≠ 2. The principal results for the problem seems to be given in (cf [1, 2, 3, 5, 6, 9, 10, 11, 12, 13]), where is shown that there exists a sequence of eigenvalues of (V.P_{(m,Ω)}) given by,

λ_{n}(m, Ω) = inf_{K∈B_{n}} max_{v∈K} \frac{∫_{Ω} |∇v|^{p} dx}{∫_{Ω} m|v|^{p} dx}  (1)

B_{n} = {K, symmetrical compact, 0 ∉ K, and γ(K) ≥ n }, γ is the genus function, or equivalently,

\frac{1}{λ_{n}(m, Ω)} = sup_{K∈B_{n}} min_{v∈K} \frac{∫_{Ω} m|v|^{p} dx}{∫_{Ω} |∇v|^{p} dx}  (2)

which can be written simply,

\frac{1}{λ_{n}(m, Ω)} = sup_{K∈A_{n}} min_{v∈K} ∫_{Ω} m|v|^{p} dx  (3)

A_{n} = {K ∩ S, K ∈ B_{n}}. S is the unit sphere of W_{0}^{1,p}(Ω) endowed with the usual norm (∥v∥_{1,p}^{p} = ∫_{Ω} |∇v|^{p} dx), the equation (2) is the generalized Rayleigh quotient for the problem (V.P_{(m,Ω)}). The sequence is ordered as 0 < λ_{1}(m, Ω) < λ_{2}(m, Ω) ≤ λ_{3}(m, Ω) ≤ ⋯ → +∞. The first eigenvalue λ_{1}(m, Ω) is of special importance. We give some of its properties which will be of interest for us (cf [1]). First, λ_{1}(m, Ω) is given by,

\frac{1}{λ_{1}(m, Ω)} = sup_{v∈S} ∫_{Ω} m|v|^{p} dx = ∫_{Ω} m|ϕ_{1}|^{p} dx  (4)

ϕ_{1} ∈ S is any eigenfunction corresponding to λ_{1}(m, Ω), for this reason λ_{1}(m, Ω) is called the principal eigenvalue, also we know that λ_{1}(m, Ω) > 0, simple (i.e if v and u are two eigenfunctions corresponding to λ_{1}(m, Ω) then v = αu for some α ∈ IR), isolate (i.e there is no eigenvalue in ]0, a[ for some a > λ_{1}(m, Ω), finally it is the unique eigenvalue which has an eigenfunction with constant sign. We denote ϕ_{1}(x) the positive eigenfunction corresponding to λ_{1}(m, Ω), ϕ_{1}(x) verifies the strong maximum principle (cf [17]), \frac{∂ϕ_{1}}{∂n} (x) < 0, for x in ∂Ω satisfying the interior ball condition.

In [14] Otani considers the case N = 1, m(x) ≡ 1, and proves that , σ_{p}(−Δ_{p}, 1) = {μ_{k}(m, I), k = 1, 2, ⋯}, the sequence can be ordered as 0 < μ_{1}(m, Ω) < μ_{2}(m, Ω) < μ_{3}(m, Ω) < ⋯ → +∞, the k-th eigenfunction has exactly k −1 zeros in I = (a, b). In [10] Elbert proved the same results in the case N = 1, m(x) ≥ 0 and continuous, the author gives an asymptotic relation of eigenvalues.

In this paper we consider the general case, N = 1 and m(x) can change sign and is not necessarily continuous. We prove that σ_{p}^{+}(−Δ_{p}, m) = {λ_{k}(m, I), k = 1, 2, ⋯}, the sequence can be ordered as 0 < λ_{1}(m, Ω) < λ_{2}(m, Ω) < λ_{3}(m, Ω) < ⋯ λ_{k}(m, I) → +∞ as k → +∞, the k-th eigenfunction has exactly k − 1 zeros in I = (a, b). The eigenvalues verify the SMP with respect to the weight m and the domain I.

In the next section we denote by: M(I) := {m ∈ L^{∞}(I)/meas{x ∈ I, m(x) > 0} ≠ 0}, m_{/J} the restriction of m on J for a subset J of I, Z(u) = {t ∈ I/ u(t) = 0}, a nodal domain ω of u is a component of I\Z(u), where (u, λ(m, I)) is a solution of (V.P_{(m,I)}). ũ_{/ω} is the extension, by zero, on I of u_{/ω}

2 Results and technical Lemmas

We first state our main results

Theorem 1 Assume that N=1 (Ω =]a, b[= I), m ∈ M(I) such that m ≢ 1 and p ≠ 2, we have

1. Every eigenfunction corresponding to the k-th eigenvalue λ_{k}(m, I), has exactly k-1 zeros.

2. For every k, λ_{k}(m, I) is simple and verifies the strict monotonicity property with respect to the weight m and the domain I.

3. σ_{p}^{+}(−Δ_{p}, m) = {λ_{k}(m, I), k = 1, 2, ⋯}, for any m ∈ M(I). The eigenvalues are ordered as 0 < λ_{1}(m, Ω) < λ_{2}(m, Ω) < λ_{3}(m, Ω) < ⋯ λ_{k}(m, I) → +∞ as k → +∞.

Corollary 1 For any integer n, we have the simple variational formulation,

\frac{1}{λ_{n}(m, I)} = sup_{F ∈F_{n}} min_{F ∩S} ∫_{a}^{b} m|v|^{p} dx  (5)

F_{n}={F / F is a n dimensional subspace of W_{0}^{1,p}(I)}.

For the proof of Theorem 1 we need some technical Lemmas.

Lemma 1 Let m, m′ ∈ M(I), m(x) ≤ m′(x), then for any n, λ_{n}(m′, I) ≤ λ_{n}(m, I)

Proof Making use of equation (2), we obtain immediately λ_{n}(m′, I) ≤ λ_{n}(m, I).

Lemma 2 Let (u, λ(m, I)) be a solution of (V.P_{(m,I)}), m ∈ M(I), then m_{/ω} ∈ M(ω) for any nodal domain ω of u.

Proof Let ω be a nodal domain of u and multiply (V.P_{(m,I)}) by ũ_{/ω} so that we obtain

0 < ∫_{ω} |u′|^{p} dx = λ(m, I) ∫_{ω} m|u|^{p} dx .  (6)

This completes the proof.

Lemma 3 The restriction of a solution (u, λ(m, I)) of problem (V.P_{(m,I)}), on a nodal domain ω, is an eigenfunction of problem (V.P_{(m_{/ω},ω)}), and we have λ(m, I) = λ_{1}(m/_{ω}, ω).

Proof Let v ∈ W_{0}^{1,p}(ω) and let ṽ be the extension by zero of v on Ω. It is obvious that ṽ ∈ W_{0}^{1,p}(Ω). Multiply (V.P_{(m,Ω)}) by ṽ

∫_{ω} |u′|^{p−2}u′v′ dx = λ(m, I) ∫_{ω} m|u|^{p−2}uv dx  (7)

for all v ∈ W_{0}^{1,p}(ω). Hence the restriction of u in ω is a solution of problem (V.P_{(m_{/ω},ω)}) with constant sign. We then have λ(m, Ω) = λ_{1}(m_{/ω}, ω), ω), which completes the proof.

Lemma 4 Each solution (u, λ(m, I)) of the problem (V.P_{(m,I)}) has a finite number of zeros.

Proof This Lemma plays an essential role in our work. We start by showing that u has a finite number of nodal domains. Assume that there exists a sequence I_{k}, k ≥ 1, of nodal domains (intervals), I_{i} ∩ I_{j} = ∅ for i ≠ j. We deduce by Lemmas 3 and 1, respectively, that

λ(m, I) = λ_{1}(m, I_{k}) ≥ λ_{1}(C, I_{k}) = \frac{λ_{1}(1, I_{k})}{C} = \frac{λ_{1}(1,]0, 1[)}{C(meas(I_{k}))^{p}} ,  (8)

where C = ∥m∥_{∞}. From equation (8) we deduce (meas(I_{k})) ≥ (\frac{λ_{1}(1,]0,1[)}{λ(m,I)C})^{\frac{1}{p}}, for all k, so

meas(I) = ∑_{k≥1}(meas(I_{k})) = +∞.

This yields a contradiction.

Let {I_{1}, I_{2}, ⋯ I_{k}} be the nodal domains of u. Put I_{i} =]a_{i}, b_{i}[, where a ≤ a_{1} < b_{1} ≤ a_{2} < b_{2} ≤ ⋯ a_{k} < b_{k} ≤ b. It is clear that the restriction of u on ]a, b_{1}[ is a nontrivial eigenfunction with constant sign corresponding to λ(m, I). The maximum principle (cf [17]) yields u(t) > 0 for all t ∈]a, b_{1}[, so a = a_{1}. By a similar argument we prove that b_{1} = a_{2}, b_{2} = a_{3}, ⋯ b_{k} = b, which completes the proof.

Lemma 5 (cf [16]) Let u be a solution of problem (V.P_{(m,Ω)}) and u ∈ W^{1,p}(Ω) ∩ L^{∞}(Ω) then u ∈ C^{1,α}(Ω) ∩ C^{1}(Ω̄) for some α ∈ (0, 1).

3 Proof of main results

Proof of Theorem 1

For n = 1, we know that λ_{1}(m, I) is simple, isolate and the corresponding eigenfunction has constant sign. Hence it has no zero in (a, b) and it remains to prove the SMP.

Proposition 1 λ_{1}(m, I) verifies the strict monotonicity property with respect to weight m and the domain I. i.e If m, m′ ∈ M(I), m(x) ≤ m′(x) and m(x) < m′(x) in some subset of I of nonzero measure then,

λ_{1}(m′, I) < λ_{1}(m, I)  (9)

and, if J is a strict sub interval of I such that m_{/J} ∈ M(J) then,

λ_{1}(m, I) < λ_{1}(m_{/J}, J).  (10)

Proof Let m, m′ ∈ M(I) as in Proposition 1 and recall that the principal eigenfunction ϕ_{1} ∈ S corresponding to λ_{1}(m, I) has no zero in I; i.e ϕ_{1}(t) ≠ 0 for all t ∈ I. By (3), we get

\frac{1}{λ_{1}(m, I)} = ∫_{I} m|ϕ_{1}|^{p} dx < ∫_{I} m′|ϕ_{1}|^{p} dx ≤ sup_{v∈S} ∫_{I} m′|v|^{p} dx = \frac{1}{λ_{1}(m′, I)}.  (11)

Then inequality (9) is proved. To prove inequality (10), let J be a strict sub interval of I and m_{/J} ∈ M(J). Let u_{1} ∈ S be the (principal) positive eigenfunction of (V.P_{(m,J)}) corresponding to λ_{1}(m_{/J}, J), and denote by ũ_{1} the extension by zero on I. Then

\frac{1}{λ_{1}(m_{/J}, J)} = ∫_{J} m|u_{1}|^{p} dx = ∫_{I} m|ũ_{1}|^{p} dx < sup_{v∈S} ∫_{I} m|v|^{p} dx = \frac{1}{λ_{1}(m, I)}.  (12)

The last strict inequality holds from the fact that ũ_{1} vanishes in I/J so can’t be an eigenfunction corresponding to the principal eigenvalue λ_{1}(m, I).

For n = 2 we start by proving that λ_{2}(m, I) has a unique zero in (a, b).

Proposition 2 There exists a unique real c_{2,1} ∈ I, for which we have Z(u) = {c_{2,1}} for any eigenfunction u corresponding to λ_{2}(m, I). For this reason, we will say that c_{2,1} is the zero of λ_{2}(m, I).

Proof Let u be an eigenfunction corresponding to λ_{2}(m, I). u changes sign in I. Consider I_{1} =]a, c[ and I_{2} =]c′, b[ two nodal domains of u, by Lemma 3, λ_{1}(m_{/I_{1}}, I_{1}) = λ_{2}(m, I) = λ_{1}(m_{/I_{2}}, I_{2}). Assume that c < c′, choose d ∈]c, c′[ and put J_{1} =]a, d[, J_{2} =]d, b[; hence J_{1} ∩ J_{2} = ∅, and for i = 1, 2, I_{i} ⊂ J_{i} strictly, and m_{/J_{i}} ∈ M(J_{i}). Making use of Lemma 3, by (10), we get

λ_{1}(m_{/J_{1}}, J_{1}) < λ_{1}(m_{/I_{1}}, I_{1}) = λ_{2}(m, I)  (13)

and

λ_{1}(m_{/J_{2}}, J_{2}) < λ_{1}(m_{/I_{2}}, I_{2}) = λ_{2}(m, I).  (14)

Let ϕ_{i} ∈ S be an eigenfunction corresponding to λ_{1}(m_{/J_{i}}, J_{i}), by (4) we have for i = 1, 2

\frac{1}{λ_{1}(m, J_{i})} = ∫_{J_{i}} m|ϕ_{i}|^{p} dx  (15)

ϕ̃_{i} is the extension by zero of ϕ_{i} on I. Consider the two dimensional subspace F = ⟨ϕ̃_{1}, ϕ̃_{2}⟩ and put K_{2} = F ∩ S ⊂ W_{0}^{1,p}(I). Obviously γ(K_{2}) = 2 and we remark that for v = αϕ̃_{1} + βϕ̃_{2}, ∥v∥_{1,p} = 1 ⟺ |α|^{p} + |β|^{p} = 1. Hence by (3), (13), (14) and (15) we obtain,

\frac{1}{λ_{2}(m,I)} ≥ min_{v∈K_{2}} ∫_{I} m|v|^{p} dx
= min_{v=αϕ̃_{1}+βϕ̃_{2}∈K_{2}}(|α|^{p} ∫_{J_{1}} m|ϕ_{1}|^{p} + |β|^{p} ∫_{J_{2}} m|ϕ_{2}|^{p})
= |α_{0}|^{p} ∫_{J_{1}} m|ϕ_{1}|^{p} + |β_{0}|^{p} ∫_{J_{2}} m|ϕ_{2}|^{p})
= \frac{|α_{0}|^{p}}{λ_{1}(m_{/J_{1}},J_{1})} + \frac{|α_{0}|^{p}}{λ_{1}(m_{/J_{2}},J_{2})}
> \frac{|α_{0}|^{p}+|β_{0}|^{p}}{λ_{2}(m,I)}
= \frac{1}{λ_{2}(m,I)},

a contradiction; hence c = c′. On the other hand, let v be another eigenfunction corresponding to λ_{2}(m, I). Denote by d its unique zero in (a, b). Assume, for example, that c < d. By Lemma 3 and relation (10), we get

λ_{2}(m, I) = λ_{1}(m_{/]a,d[},]a, d[) < λ_{1}(m_{/]a,c[},]a, c[) = λ_{2}(m, I).  (16)

This is a contradiction so c = d. We have proved that every eigenfunction corresponding to λ_{2}(m, I) has one, and only one, zero in (a, b), and that the zero is the same for all eigenfunctions, which completes the proof of the Proposition.

Lemma 6 λ_{2}(m, I) is simple, hence λ_{2}(m, I) < λ_{3}(m, I).

Proof Let u and v be two eigenfunctions corresponding to λ_{2}(m, I). By Lemma 3 the restrictions of u and v on ]a, c_{2,1}[ and ]c_{2,1}, b[ are eigenfunctions corresponding to λ_{1}(m_{/]a,c_{2,1}[},]a, c_{2,1}[) and λ_{1}(m_{/]c_{2,1},b[},]c_{2,1}, b[), respectively. Making use of the simplicity of the first eigenvalue, we get u = αv in ]a, c_{2,1}[ and u = βv in ]c_{2,1}, b[. But both of u and v are eigenfunctions, so then by Lemma 5, there are C^{1}(I). The maximum principle (cf [17]) tell us that u′(c_{2,1}) ≠ 0, so α = β. Finally, by the simplicity of λ_{2}(m, I) and the theorem of multiplicity (cf [15]) we conclude that λ_{2}(m, I) < λ_{3}(m, I).

Proposition 3 λ_{2}(m, I) verifies the SMP with respect to the weight m and the domain I.

Proof Let m, m′ ∈ M(I) such that, m(x) ≤ m′(x) a.e in I and m(x) < m′(x) in some subset of nonzero measure. Let c_{2,1} and c′_{2,1} be the zeros of λ_{2}(m, I) and λ_{2}(m′, I) respectively. We distinguish three cases :

1. c_{2,1} = c′_{2,1} = c, then meas({x ∈ I/ m(x) < m′(x)∩]a, c[}) ≠ 0, or meas({x ∈ I/ m(x) < m′(x)∩]c, b[}) ≠ 0, by Lemma 3 and (9) we obtain

λ_{2}(m′, I) = λ_{1}(m′_{/]a,c[},]a, c[) < λ_{1}(m_{/]a,c[},]a, c[) = λ_{2}(m, I),  (17)

or

λ_{2}(m′, I) = λ_{1}(m′_{/]c,b[},]c, b[) < λ_{1}(m_{/]c,b[},]c, b[) = λ_{2}(m, I).  (18)

2. c_{2,1} < c′_{2,1}, by Lemmas 1, 3 and (10), we get

\begin{aligned}λ_{2}(m′, I) = λ_{1}(m′_{/]a,c′_{2,1}[},]a, c′_{2,1}[) & ≤ & λ_{1}(m_{/]a,c′_{2,1}[},]a, c′_{2,1}[) \\ & < & λ_{1}(m_{/]a,c_{2,1}[},]a, c_{2,1}[) = λ_{2}(m, I). \end{aligned}  (19)

3. c′_{2,1} < c_{2,1}, as before, by Lemmas 1, 3 and (10), we have

\begin{aligned}λ_{2}(m′, I) = λ_{1}(m′_{/]c′_{2,1},b[},]c′_{2,1}, b[) & ≤ & λ_{1}(m_{/]c′_{2,1},b[},]c′_{2,1}, b[) \\ & < & λ_{1}(m_{/]c_{2,1},b[},]c_{2,1}, b[) = λ_{2}(m, I). \end{aligned}  (20)

For the SMP with respect to the domain, put J =]c, d[ a strict sub interval of I with m_{/J} ∈ M(J), and denote c′_{2,1} the zero of λ_{2}(m_{/J}, J). As in the SMP with respect to the weight, three cases are distinguished:

1. c_{2,1} = c′_{2,1} = l, then ]c, l[ is a strict sub interval of ]a, l[ or ]l, d[ is a strict sub interval of ]l, b[. By Lemma 3 and (10), we get

λ_{2}(m, I) = λ_{1}(m_{/]a,l[},]a, l[) < λ_{1}(m_{/]c,l[},]c, l[) = λ_{2}(m_{/J}, J)  (21)

or

λ_{2}(m, I) = λ_{1}(m_{/]l,b[},]l, b[) < λ_{1}(m_{/]l,d[},]l, d[) = λ_{2}(m_{/J}, J).  (22)

2. c_{2,1} < c′_{2,1}, again by Lemma 3 and (10), we obtain

λ_{2}(m, I) = λ_{1}(m_{/]c_{2,1},b[},]c_{2,1}, b[) < λ_{1}(m_{/]c′_{2,1},d[},]c′_{2,1}, b[) = λ_{2}(m_{/J}, J).  (23)

3. c′_{2,1} < c_{2,1}, for the same reason as in the last case, we get

λ_{2}(m, I) = λ_{1}(m_{/]a,c_{2,1}[},]a, c_{2,1}[) < λ_{1}(m_{/]c,c′_{2,1}[},]c, c_{2,1}[) = λ_{2}(m_{/J}, J).  (24)

The proof is complete.

Lemma 7 If any eigenfunction u corresponding to some eigenvalue λ(m, I) is such that Z(u) = {c} for some real number c, then λ(m, I) = λ_{2}(m, I).

Proof We shall prove that c = c_{2,1}. Assume, for example, that c < c_{2,1}. By Lemma 1 and (10) we get

λ(m, I) = λ_{1}(m_{/]c,b[},]c, b[) < λ_{1}(m_{/]c_{2,1},b[},]c_{2,1}, b[) = λ_{2}(m, I).  (25)

On the other hand,

λ_{2}(m, I) = λ_{1}(m_{/]a,c_{2,1}[},]a, c_{2,1}[) < λ_{1}(m_{/]a,c[},]a, c[) = λ(m, I),  (26)

a contradiction. Hence, c = c_{2,1} and λ(m, I) = λ_{2}(m, I). The proof is complete.

For n > 2, we use a recurrence argument. Assume that, for any k, 1 ≤ k ≤ n, that the following hypothesis:

1. H.R.1 For any eigenfunction u corresponding to the k-th eigenvalue λ_{k}(m, I), there exists a unique c_{k,i}, 1 ≤ i ≤ k − 1, such that Z(u) = {c_{k,i}, 1 ≤ i ≤ k − 1}.

2. H.R.2 λ_{k}(m, I) is simple.

3. H.R.3 λ_{1}(m, I) < λ_{2}(m, I) < ⋯ < λ_{n+1}(m, I).

4. H.R.4 If (u, λ(m, I)) is a solution of (V.P_{(m,I)}) such that Z(u) = {c_{i}, 1 ≤ i ≤ k−1}, then λ(m, I) = λ_{k}(m, I).

5. H.R.5 λ_{k}(m, I) verifies the SMP with respect to the weight m and the domain I.

Holds, and prove them for n + 1.

Proposition 4 There exists a unique family {c_{n+1,i}, 1 ≤ i ≤ n} such that Z(u) = {c_{n+1,i}, 1 ≤ i ≤ n}, for any eigenfunction u corresponding to λ_{n+1}(m, I).

Proof Let u be an eigenfunction corresponding to λ_{n+1}(m, I). By H.R.3 and H.R.4, u has at least n zeros. According to Lemma 4, we can consider the n + 1 nodal domains of u, I_{1} =]a, c_{1}[, I_{2} =]c_{1}, c_{2}[, ... , I_{n} =]c_{n−1}, c_{n}[, I_{n+1} =]c, b[. We shall prove that c = c_{n}. Remark that the restrictions of u on ]a, c_{i+1}[, 1 ≤ i ≤ n − 1, are eigenfunctions with i zeros, by H.R.4 λ_{n+1}(m, I) = λ_{i}(m_{/]a,c_{i+1}[},]a, c_{i+1}[). Assume that c_{n} < c, choose d in ]c_{n}, c[ and put, J_{1} =]a, d[, J_{2} =]d, b[. Remark that J_{1} ∩ J_{2} = ∅, ]a, c_{n}[ is a strict sub interval of J_{1} ⊂ I, and ]c, b[ is a strict sub interval of J_{2} ⊂ I. It is clear that m_{/J_{i}} ∈ M(J_{i}) for i = 1, 2, by H.R.4 and H.R.5. We have

λ_{n}(m_{/J_{1}}, J_{1}) < λ_{n}(m_{/]a,c_{n}[},]a, c_{n}[) = λ_{n+1}(m, I),  (27)

and

λ_{1}(m_{/J_{2}}, J_{2}) < λ_{1}(m_{/]c,b[},]c, b[) = λ_{n+1}(m, I).  (28)

Denote by (ϕ_{n+1}, λ_{1}(m_{/J_{2}}, J_{2})) a solution of (V.P_{(m,J_{2})}), (v, λ_{n}(m_{/J_{1}}, J_{1})) a solution of (V.P_{(m,J_{1})}), ϕ_{i}, 1 ≤ i ≤ n the restrictions of v on I_{i}, and ϕ̃_{i}, 1 ≤ i ≤ n, their extensions, by zero, on I. Let F_{n+1} = ⟨ϕ̃_{1}, ϕ̃_{2}, ⋯, ϕ̃_{n+1}⟩ and K_{n+1} = F_{n+1} ∩ S, then γ(K_{n+1}) = n + 1. We obtain by (3) and the same proof as in Proposition 2

\frac{1}{λ_{n+1}(m, I)} ≥ min_{K_{n+1}} ∫_{I} m|v|^{p} dx > \frac{1}{λ_{n+1}(m, I)},  (29)

a contradiction, so c = c_{n}. On the other hand, let v be an eigenfunction corresponding to λ_{n+1}(m, I). Denote by d_{1}, d_{2}, ⋯, d_{n} the zeros of v. If d_{1} ≠ c_{1}, then λ_{n+1}(m, I) = λ_{1}(m_{/]a,d_{1}[},]a, d_{1}[) ≠ λ_{1}(m_{/]a,c_{1}[},]a, c_{1}[) = λ_{n+1}(m, I), so d_{1} = c_{1}, by the same argument we conclude that d_{i} = c_{i} for all 1 ≤ i ≤ n.

Lemma 8 λ_{n+1}(m, I) is simple, hence. λ_{n+1}(m, I) < λ_{n+2}(m, I).

Proof Let u and v be two eigenfunctions corresponding to λ_{n+1}(m, I). The restrictions of u and v on ]a, c_{n+1,1}[ and ]c_{n+1,1}, b[ respectively, are eigenfunctions corresponding to λ_{1}(m_{/]a,c_{n+1,1}[},]a, c_{n+1,1}[) and λ_{n}(m_{/]c_{n+1,1},b[},]c_{n+1,1}, b[). By H.R.2 and H.R.4 we have u = αv in ]a, c_{n+1,1}[ and u = βv in ]c_{n+1,1}, b[. On the other hand, u and v are C^{1}(I) and u′(c_{n+1,1}) ≠ 0, so α = β. From the simplicity of λ_{n+1}(m, I) and theorem of multiplicity we conclude that λ_{n+1}(m, I) < λ_{n+2}(m, I).

Proposition 5 λ_{n+1}(m, I) verifies the SMP with respect to the weight m and the domain I.

Proof Let m, m′ ∈ M(I), such that m(x) ≤ m′(x) with m(x) < m′(x) in some subset of nonzero measure and (cn+1,i^{′})_{1≤i≤n} the zeros of λ_{n+1}(m′) three cases are distinguished,

1. c_{n+1,1} = c′_{n+1,1} = c, one of the subsets is of nonzero measure,

{x ∈ I/ m(x) < m′(x)}∩]a, c[ and {x ∈ I/ m(x) < m′(x)}∩]c, b[.

By Lemma 3 and (9), we have

λ_{n+1}(m′, I) = λ_{1}(m′_{/]a,c[},]a, c[) < λ_{1}(m_{/]a,c[},]a, c[) = λ_{n+1}(m, I)  (30)

or

λ_{n+1}(m′, I) = λ_{n}(m′_{/]c,b[},]c, b[) < λ_{n}(m_{/]c,b[},]c, b[) = λ_{n+1}(m, I).  (31)

2. c_{n+1,1} < c′_{n+1,1}, by Lemmas 1, 3 and (10) we have

\begin{aligned}λ_{n+1}(m′, I) & = & λ_{1}(m′_{/]a,c′_{n+1,1}[},]a, c′_{n+1,1}[) \\ & ≤ & λ_{1}(m_{/]a,c′_{n+1,1}[},]a, c′_{n+1,1}[) \\ & < & λ_{1}(m_{/]a,c_{n+1,1}[},]a, c_{n+1,1}[) & = & λ_{n+1}(m, I). \end{aligned}  (32)

3. c′_{n,1} < c_{n,1}, from the same reason as before, we get

\begin{aligned}λ_{n+1}(m′, I) & = & λ_{n}(m′_{/]c′_{n+1,1},b[},]c′_{n+1,1}, b[) \\ & ≤ & λ_{n}(m_{/]c′_{n+1,1},b[},]c′_{n+1,1}[, b) \\ & < & λ_{n}(m_{/]c_{n+1,1},b[},]c_{n+1,1}, b[) & = & λ_{n+1}(m, I). \end{aligned}  (33)

By similar argument as in proof of Proposition 3, we prove the SMP with respect to the domain I.

Lemma 9 If (u, λ(m, I)) is a solution of (V.P_{(m,I)}) such that Z(u) = {d_{1}, d_{2}, ⋯ d_{n}}, then λ(m, I) = λ_{n+1}(m, I).

Proof It is sufficient to prove that d_{i} = c_{n+1,i} for all 1 ≤ i ≤ n. If c_{n+1,1} < d_{1} then, by Lemma 1, (10), H.R.4 and H.R.5,

\begin{aligned}λ(m, I) = λ_{1}(m_{/]a,d_{1}[},]a, d_{1}[) & < & λ_{1}(m_{/]a,c_{n+1,1}[},]a, c_{n+1,1}[) \\ & = & λ_{n+1}(m, I) \\ & = & λ_{n}(m_{/]c_{n+1,1},b[},]c_{n+1,1}, b[) \\ & < & λ_{n}(m_{/]d_{1},b[},]d_{1}, b[) \\ & = & λ(m, I), \end{aligned}  (34)

a contradiction. If d_{1} < c_{n+1,1} then, by Lemma 1, (10), H.R.4 and H.R.5,

\begin{aligned}λ_{n+1}(m, I) = λ_{1}(m_{/]a,c_{n+1}[},]a, c_{n+1}[) & < & λ_{1}(m_{/]a,d_{1}[},]a, d_{1}[) \\ & = & λ(m, I) \\ & = & λ_{n}(m_{/]d_{1},b[},]d_{1}, b[) \\ & < & λ_{n}(m_{/]c_{n+1,1},b[},]c_{n+1}, b[) \\ & = & λ_{n+1}(m, I), \end{aligned}  (35)

a contradiction. The proof is then complete, which completes the proof of Theorem 1.

Proof of Corollary 1. Since for F ∈ F_{n}, the compact F ∩ S ∈ A_{n}, by (3) we have:

sup_{F ∈F_{n}} min_{v∈F ∩S} ∫_{Ω} m|v|^{p} dx ≤ \frac{1}{λ_{n}(m, Ω)}.  (36)

On the other hand, for a n dimensional subspace F of W_{0}^{1,p}(I), the compact set K = F ∩ S ∈ A_{n}. Let u be an eigenfunction corresponding to λ_{n}(m, I) and put

F = ⟨ϕ̃_{1}(]a, c_{n,1}[), ϕ̃_{1}(]c_{n,1}, c_{n,2}[), ⋯, ϕ̃_{1}(]c_{n_{1},n}, b[)⟩,

to conclude F ∩ S ∈ A_{n}. By an elementary computation as in Proposition 2, one can show that

\frac{1}{λ_{n}(m, I)} = min_{F ∩S} ∫_{I} m|v|^{p} dx.  (37)

Then combine (36) with (37) to get (5). Which completes the proof.

3.1 Remark

The spectrum of p-Laplacian, with indefinite weight, in one dimension, is entirely determined by the sequence (λ_{n}(m, I))_{n≥1} if m(x) ≥ 0 a.e in I. Therefore, if m(x) < 0 in some subset J ⊂ I of nonzero measure, replace m by −m; by Theorem 1, since −m ∈ M(I) we conclude that, the negative spectrum σ_{p}^{−}(−Δ_{p}, m) = −σ_{p}^{+}(−Δ_{p}, −m) of this operator is constituted by a sequence of eigenvalues λ_{−n}(m, I) = −λ_{n}(−m, I). Thus the spectrum of the operator is,

σ_{p}(−Δ_{p}, m) = σ_{p}^{+}(−Δ_{p}, m) ∪ σ_{p}^{−}(−Δ_{p}, m).

References

[1] A. Anane, Simplicité et isolation de la première valeur propre du p-Laplacien avec poids, C. R. Acad. Sci. Paris, t.305, Série 1, (1987), pp. 725-728.

[2] A. Anane and O. Chakrone, Sur un théorème de point critique et application à un problème de non résonance entre deux valeurs propres du p-Laplacien, Annales de la Faculté des Sciences de Toulouse, Vol. IX, No.1, (2000), pp. 5-30.

[3] A. Anane and N. Tsouli, On the second eigenvalue of the p-Laplacian, Nonlinear Partial Differential Equations, Benkirane A., Gossez J.P., (Eds.), Pitman Res. Notes in Math. 343, (1996), pp. 1-9.

[4] R. Courant and D. Hilbert, Methods of Mathematical Physics, Intersience, New York, (1943).

[5] M. Cuesta, D.G. De Figueiredo and J.P. Gossez, The beginning of the Fucik spectrum for the p-Laplacian, J. Differential Equations, 159, (1999), pp. 212-238.

[6] M. Cuesta, D.G. De Figueiredo and J.P. Gossez, A nodal domain property for the p-Laplacian, C. R. Acad. Sci. Paris, t.330, Série I (2000), pp. 669-673.

[7] D.G. De Figueiredo, Positive solution of semi-linear elliptic equation, Lecture Notes in Mathematics, 957, (1982), pp. 34-87.

[8] D.G. De Figueiredo, and J.P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Part. Diff. Equat., 17, (1992) pp. 339-346.

[9] M. Del Pino, M. Elgueta and R. Manàsevich, A Homotopic deformation along p of a Leray-Schauder degree result and existence for (|u′|^{p−2}u′)′ + f(t, u) = 0, u(0) = u(T) = 0, p > 1, J. Differential Equations, 80, (1989), 1-13.

[10] A. Elbert, A half-linear second order differential equation, Colloquia Mathematica Societatis Janos Bolyai 30. Qualitative Theory of Differential Equations, Szeged, (Hungary) (1979), pp 124-143.

[11] S. F učik, J. Nečas and V. Souček, Spectral analysis of nonlinear operators, Lect. Notes Math. 346, Springer Verlag, New York. Berlin. Heidelberg (1973).

[12] Y.X. Huang, Eigenvalues of the p-Laplacian in IR^{N}with indefinite weight, Comment. Math. Univ. Carolinae. (1995), pp 519-527.

[13] P. Lindqvist, On the equation div(|∇u|^{p−2}∇u) + λ|u|^{p−2}u = 0, Proc. Amer. Math. Soc. 109 (1990), pp. 157-164; addendum Proc. Amer. Math. Soc. 116, (1992), pp. 583-584.

[14] M. Otani, A remark on certain nonlinear elliptic equations, Pro. Fac. Sci. Tokai Univ. No.19 (1984), pp. 23-28.

[15] A. Szulkin, Ljusternik-Schnirelmann theory on C^{1}-manifolds, Ann. I. H. Poincaré, Anal. non linéaire, 5, (1988), pp. 119-139.

[16] P. Tolksdorf, Regularity for more general class of quasilinear elliptic equations, J. Differential Equations, 51, (1984), pp. 126-150.

[17] J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12, (1984), pp. 191-202.